• Ph.D. candidate in Department of Political Science
• Research area:
  • Comparative politics, methodology, and international relations.
  • Substantive: language policy, political communication, and public opinion.
  • Methodology:
    • Survey experiments
    • Quantitative methods
• Homepage: https://sammo3182.github.io/.

• Today
  • Functions
• Tomorrow
  • Linear algebra (10:00 - 12:00)
  • Calculus:　Differential (13:30 - 15:30)

• Definition: "A relation that assigns one element of the range to each element of the domain (Moore & Siegel 2013, 45)"
  • \(Y = f(X).\)
  • X: domain
  • Y: codomain

• Domain:
  
  
  
• Codomain:
  
  
  

• Function:
  
  
  

• Equation:
  • Implicit function: \(y = f(X) = 1 - x^2.\)
    • y is a function of x
      • Image/range.
    • x: Argument.
    • f: Mapping.
      • We map the values of Y for any given set of X values.
      • f maps A into B.
  • Explicit function: \(y = 1 - x^2.\)
  • Function composition: \(g(f(x))\), "g composed with f."
• Graph: https://www.desmos.com/calculator

• A function is surjective or onto if every value in the codomain is produced by some value in the domain.
• A function is injective or one-to-one if each value in the range comes from only one value in the domain.
• If a function is both injective and surjective (one-to-one and onto), then it is bijective.
  • A bijective function is invertible, viz. having an inverse.

• Identity function: \(f(x) = x.\)

• Inverse function: \[ \begin{align} f(x) =& 2x + 3;\\ f^{-1}(x) =& \frac{x - 3}{2}. \end{align} \]

• \(f(f^{-1}(x)) = ?\)
  • \(x\).
• \(f^{-1}(f(x)) = ?\)
  • \(x\).

• Increasing vs. Decreasing
• Strictly increasing vs. strictly decreasing
  • "A strictly monotonic function is strictly increasing over its entire domain (Moore & Siegel 2013, 50)."
  • A function is monotonic if its first derivative does not change sign.
• Weakly increasing vs. Weakly decreasing
  • Not decreasing
  • Not increasing

\[Y = \beta_1Weight + \beta_2Cylinders + \beta_3Weight\times Cylinders.\]

\[y = ax + b.\] * a: slope * b: intercept. * Properties: + Additivity (superposition): \(f(x_1 + x_2) = f(x_1) + f(x_2).\) + Scaling (homogeneity): \(f(ax) = af(x).\)

\[b^n = x.\]

• Exponential function is to solve for x.
  • e.g., \(y = x^2.\)
• Property: x is called the base.
  • Multiplication: m and n are constant.
    • \(x^m + x^n = x^{m + n};\)
    • \(x^m * z^m = (xz)^m;\)
    • \((x^m)^n = x^{mn}.\)
    • \(x^0 = 1.\)

• Division
  • \(\frac{x^m}{x^n} = x^{m - n};\)
    • So what's \(\frac{1}{x^n}\)?
    • \(x^{-n}.\)
  • \(\frac{x^m}{z^m} = (\frac{x}{z})^m.\)
• Some terminology
  • Quadratic (parabola): \(y = \beta_0 + \beta_1x + \beta_2x^2.\)
  • Polynomial: \[\begin{align} y =& \beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3 + ...\\ y =& \sum_0^n\beta_nx^n. \end{align}\]

\[b^n = x.\]

• Radical function is to solve for b.
  • e.g., \(y = \sqrt[n]{x} = x^{\frac{1}{n}}.\)
• Properties
  • Multiplication: \(\sqrt[n]{x}\times \sqrt[n]{z} = x^{\frac{1}{n}}\times z^{\frac{1}{n}} = (xz)^{\frac{1}{n}}.\)
  • Division: \(\frac{\sqrt[n]{x}}{\sqrt[n]{z}} = (\frac{x}{z})^{\frac{1}{n}}.\)

\[b^n = x.\]

• Logarithm function is to solve for n.
  • The inverse of exponential function (e.g., \(y = e^x; y = 10^x\)).
  • e.g., \(y = ln(x); y = log(x).\)
• Properties
  • \(ln(1) = 0; log_a(a) = 1\)
  • \(ln(x_1\cdot x_2) = ln(x_1) + ln(x_2);\)
  • \(ln(\frac{x_1}{x_2}) = ln(x_1) - ln(x_2);\)
    • NB: \(ln(x_1\pm x_2)\neq ln(x_1)\pm ln(x_2).\)
  • \(ln(x^b) = bln(x).\)

• Correspondence: "A relation that assigns a subset of the range to each element of the domain (Moore & Siegel 2013, 45)" (?)
• In general mathematics, a correspondence is an ordered triple (X,Y,R), where R is a relation from X to Y, i.e. any subset of the Cartesian product X×Y.

1. \(x^{-2}\times x^3 = ?\)
2. \((b\cdot b\cdot b)\times c^{-3} = ?\)
3. \(((qr)^\gamma)\delta = ?\)
4. \(\sqrt{x}\times \sqrt[5]{x} = ?\)
5. \(ln(3x) - 2ln(x + 2) = ?\)
6. \[\begin{align} f(x) =& x^2 + 2;\\ g(x) =& \sqrt{x - 4}.\\ h(x) =& g(f(x)) = ? \end{align}\]

1. \(x^{-2}\times x^3 = x.\)
2. \((b\cdot b\cdot b)\times c^{-3} = b^3c^{-3}.\)
3. \(((qr)^\gamma)^\delta = (qr)^{\gamma\delta}.\)
4. \(\sqrt{x}\times \sqrt[5]{x} = x^{\frac{7}{10}}.\)
5. \(ln(3x) - 2ln(x + 2) = ln(\frac{3x}{(x + 2)^2}).\)
6. \[\begin{align} f(x) =& x^2 + 2;\\ g(x) =& \sqrt{x - 4}.\\ h(x) =& g(f(x)) = \sqrt{x^2 - 2}. \end{align}\]

1. \(f_1(x) = x + 0.5.\)
2. \(f_2(x) = -\frac{x}{3} + \frac{1}{4}.\)

1. \(y = \beta_0 + x_1^{\beta_1} + \beta_2x_2 + \beta_3x_3.\)
2. \(y = \beta_0\times x_1^{\beta_1}\times x_2^{\beta_2}\times x_3^{\beta_3}.\)
3. \(y = \beta_0\times x_1^{\beta_1}\times \frac{x_2^{\beta_2}}{x_3^{\beta_3}}.\)

1. \(ln(y) = ln(\beta_0 + x_1^{\beta_1} + \beta_2x_2 + \beta_3x_3).\)
2. \(ln(y) = 1 + \beta_1ln(x1) + \beta_2ln(x2) + \beta_3ln(x3);\)
3. \(ln(y) = 1 + \beta_1ln(x1) + \beta_2ln(x2) - \beta_3ln(x3).\)

• As a methodologist,
  • Reading the method papers.
  • Spatial analyses, game theory, dynamic model, etc.
• As an empiricist,
  • Learning R.
• As a lazy guy,
  • \(y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_nX_n + \varepsilon\)
  • \(\boldsymbol{Y} = \boldsymbol{X\beta}\)
  • And imagine if you have more than one \(y\).

• Scalar
  • 
  • Single element of a set.

• Vector
  • 
  • Force in physics
  • Dimension

• Matrix
  • 

• Length

  \[||\boldsymbol{x}|| = \sqrt{\sum x_i^2}. \]

• What's this in the real life?
  • right triangle.

• Example

  If \(\boldsymbol{x} = (2, 4, 4, 1)\), its length \(||\boldsymbol{x}|| = \sqrt{2^2 + 4^2 + 4^2 + 1^2} = \sqrt{37}\).

• Addition

  \[\boldsymbol{a} \pm \boldsymbol{b} = (a_1 \pm b_1, a_2 \pm b_2, ..., a_n \pm b_n).\]

• What's this in the real life?
  • 

• Example: \((1,2) + (5,8) = (1 + 5, 2 + 8) = (6, 10).\)

• Scalar Multiplication
  • \[c\boldsymbol{x} = (cx_1, cx_2,..., cx_n). \]
• What's this in the real life?
  • 
• Example: Let \(\boldsymbol{x} = (2, 4), 2\boldsymbol{x} = (4,8).\)

• Vector normalization
  • \[\frac{\boldsymbol{x}}{||\boldsymbol{x}||}\]
• Vector Multiplication
  • Dot product: \[\boldsymbol{a\cdot b} = \sum a_ib_i = |\boldsymbol{a}||\boldsymbol{b}|cos\theta. \]

• Column vector: a matrix with only one column
• Row vector: a matrix with only one row
• Scalar: a matrix with only one element
• Square matrix: \(A_{2\times2} = \left(\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right)\)
• Zero matrix: \(\left(\begin{array}{cc} 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)\)
• Diagonal matrix: \(\left(\begin{array}{cc} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{array}\right)\)

• Identity matrix : \(I = \left(\begin{array}{cc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right)\)

• Permutation matrix : \(\left(\begin{array}{cc} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{array}\right)\)

• Upper/lower matrix: \(\left(\begin{array}{cc} 1 & 4 & 5\\ 0 & 2 & 6\\ 0 & 0 & 3 \end{array}\right)\)

• \(A \pm B = a_{ij} \pm b_{ij}\)
• Example
  • \(A = \left(\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right)\)
  • \(B = \left(\begin{array}{cc} 5 & 6\\ 7 & 8 \end{array}\right)\)
    
  • \(A + B = \left(\begin{array}{cc} 6 & 8\\ 10 & 12 \end{array}\right)\)

• Scalar Multiplication: \(cA = ca_{ij}.\)
• Matrix Multiplication: \(C_{n\times p} = A_{n\times m}B_{m\times p} = \sum_{k = i}^{m}a_{ik}b_{kj}.\)
  • 
  • Left-multyplying B by A means \(BA\); right-multiplying B by A means \(AB\).

• Kronecker Product: \(A_{m\times n}\otimes B_{p\times q} = C_{mp\times nq}.\)
  • 

• Used to convert the matrix into scalar.
• \(2\times 2\) matrix (butterfly method)
  • 
• Laplace expansion: using minors.
  • 
• Matrices with non-zero determinant are non-singular matrices, which means invertible.

• If \(AB = BA = I\), then B is the inverse of A, i.e., \(B = A^{-1}\). Therefore, \[A\cdot A^{-1} = I.\]
• Calculate the inverse
  • Cofactor Matrix: \(A_{ij} = (-1)^{i + j}M_{ij}\), where \(M\) is the minor of \(a_{ij}\).
  • Adjoint matrix: the matrix of cofactor matrices.
  • \(A^{-1} = \frac{1}{|A|}adj(A)\)
  • Example: Moore & Siegel 2013, pp.295-6.

• Matrix
  • Associative: \((AB)C = A(BC)\).
  • Additive distributive: \((A + B)C = AB + BC\).
  • Scalar commutative: \(xAB = (xA)B = A(xB) = ABx\)
• Transpose
  • Inverse: \((A^T)^T = A\).
  • Additive: \((A + B)^T = A^T + B^T\).
  • Multiplicative: \((AB)^T = B^TA^T\).
  • Scalar multiplication: \((cA)^T = cA^T\).
  • Inverse transpose: \((A^{-1})^T = (A^T)^{-1}\).

• Determinant
  • Transpose: \(det(A) = det(A^T)\).
  • Identity: \(det(I) = 1\).
  • Multiplicative: \(det(AB) = det(A)det(B)\).
  • Inverse: \(det(A^{-1}) = \frac{1}{det(A)}\).
  • Scalar multiplicative: \(det(cA_{n\times n}) = c^ndet(A)\).
• Inverse
  • Inverse: \((A^{-1})^{-1} = A\).
  • Multiplicative: \((AB)^{-1} = B^{-1}A^{-1}\).
  • Scalar multiplicative: \((cA)^{-1} = c^{-1}A^{-1}, \mbox{if}\ c\neq 0\).

library(dplyr)
a <- matrix(c(10, 2, 5, 2), ncol = 1)
b <- matrix(c(4, 15, 6, 8), ncol = 1)
c <- matrix(c(2, 6, 8), nrow = 1)
d <- matrix(c(1, 15, 12), nrow = 1)
e <- matrix(c(14, 17, 17, 11, 10), nrow = 1) %>% t
f <- matrix(c(20, 4, 10, 4), nrow = 1)

a + b
a + c
b - e
15 * c
-3 * f
norm(b, type = "F")
norm(c + d, type = "F")
norm(c - d, type = "F")
a %*% b
c %*% d

Answers

a + b

##      [,1]
## [1,]   14
## [2,]   17
## [3,]   11
## [4,]   10

#a + c
#b - e
15 * c

##      [,1] [,2] [,3]
## [1,]   30   90  120

-3 * f

##      [,1] [,2] [,3] [,4]
## [1,]  -60  -12  -30  -12

norm(b, type = "F")

## [1] 18.46619

norm(c + d, type = "F")

## [1] 29.15476

norm(c - d, type = "F")

## [1] 9.899495

#a %*% b
#c %*% d

A <- matrix(c(5, 1, 2, 6, 2, 3), nrow = 2)
B <- matrix(c(3, 4, 5, -2, -3, 6), nrow = 2)
C <- matrix(c(1, 2, -5, 3, -3, 1), ncol = 2)
D <- matrix(c(2, 1, 4, 3), nrow = 2)

2 * B - 5 * A
t(B) - C
B %*% C
C %*% B

Answers:

2 * B - 5 * A

##      [,1] [,2] [,3]
## [1,]  -19    0  -16
## [2,]    3  -34   -3

t(B) - C

##      [,1] [,2]
## [1,]    2    1
## [2,]    3    1
## [3,]    2    5

B %*% C

##      [,1] [,2]
## [1,]   28   -9
## [2,]  -30   24

C %*% B

##      [,1] [,2] [,3]
## [1,]   15   -1   15
## [2,]   -6   16  -24
## [3,]  -11  -27   21

A <- matrix(c(2, 1, -2, 2), ncol = 2)
B <- matrix(c(3, 2, -4, -1, -5, 1, 3, 2, 3), ncol = 3)

det(A)
det(B)

Answers:

det(A)

## [1] 6

det(B)

## [1] -91

A <- matrix(c(4, 2, 6, 3), ncol = 2)
B <- matrix(c(1, 4, 3, 2), ncol = 2)

solve(A)
solve(B)

Answers

# solve(A)
solve(B)

##      [,1] [,2]
## [1,] -0.2  0.3
## [2,]  0.4 -0.1

1/det(B) * matrix(c(2, -3, -4, 1), nrow = 2)

##      [,1] [,2]
## [1,] -0.2  0.4
## [2,]  0.3 -0.1

• Mathematics definition
  • 
• what's this in the geometric?
  • 

• What's this in the real life?
  • 

• \((C)' = 0\).
• \((x^{\mu})' = \mu x^{\mu - 1}\).
• \((a^x)' = a^xln\ a\).
• \((log_ax)' = \frac{1}{xln\ a}\).
• Let u, v are derivable,
  • \((u\pm v)' = u' \pm v'\).
  • \((Cu)' = Cu'\), where \(C\) is constant.
  • \((uv)' = u'v + uv'\).
  • \((\frac{u}{v})' = \frac{u'v - uv'}{v^2},\) where \(v\neq 0.\)

• Chain rule: \(g(f(x))' = g'(f(x))f'(x)\)
  • To solve \(g(f(x))'\), let \(u = f(x)\).
  • \(g(f(x))' = g(u)'u' = g(u)'f(x)'\)